A slip coach, in British railway terminology, is a coach (in American parlance, a passenger car) that can be uncoupled from a moving train at speed and then coasted to a stop at a station under the control of a guard (US: conductor) with control of the coach's brakes.
The purpose of this is to allow an express train to drop off passengers en route without having to stop at intermediate stations. Obviously picking up passengers is not possible!
Use of a slip coach obviously requires a shunter (US: switcher) to be on standby to move the slip coach out of the way. In addition, arrangements need to be made for the slip coach to make its way back to the terminus for its next use. This complexity, and the requirement for a shunter, its crew and an extra guard, mean that it's just not cost-effective in the modern age. I do not believe any slip coaches are in use, certainly not in Britain and possibly not anywhere.
On railway systems not influenced by British practice, the idea of such an operation being normal behavior seems to be rather shocking and considered quite unsafe, but in practice (especially given the fact that the whole operation likely took place in view of the signalman) it proved not to be.